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The monolithic integration of single-photon sources, passive optical circuits, and single-photon detectors enables complex and scalable quantum photonic integrated circuits, for application in linear-optics quantum computing and quantum communications. Here, we demonstrate a key component of such a circuit, a waveguide single-photon detector. Our detectors, based on superconducting nanowires on GaAs ridge waveguides, provide high efficiency (∼20%) at telecom wavelengths, high timing accuracy (∼60 ps), and response time in the ns range and are fully compatible with the integration of single-photon sources, passive networks, and modulators.